cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A247321 Rectangular array read upwards by columns: T = T(n,k) = number of paths from (0,0) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 2, 1, 2, 4, 2, 2, 5, 5, 6, 5, 7, 13, 10, 7, 18, 22, 20, 18, 29, 45, 40, 29, 63, 87, 74, 63, 116, 166, 150, 116, 229, 329, 282, 229, 445, 627, 558, 445, 856, 1232, 1072, 856, 1677, 2373, 2088, 1677, 3229, 4621, 4050, 3229
Offset: 0

Views

Author

Clark Kimberling, Sep 13 2014

Keywords

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = k, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n.

Examples

			First 10 columns:
0 .. 0 .. 2 .. 2 .. 6 .. 10 .. 20 .. 40 .. 74 .. 150
0 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 .. 166
0 .. 1 .. 1 .. 2 .. 5 .. 7 ... 18 .. 29 .. 63 .. 116
1 .. 0 .. 1 .. 1 .. 2 .. 5 ... 7 ... 18 .. 29 .. 63
T(3,2) counts these 4 paths, given as vector sums applied to (0,0):
(1,2) + (1,1) + (1, -1)
(1,1) + (1,2) + (1,-1)
(1,2) + (1,-1) + (1,1)
(1,1) + (1,-1) + (1,2)
Partial sums of second components in each vector sum give the 3 integer strings described in Comments:  (0,2,3,2), (0,1,3,2), (0,2,1,2), (0,1,0,2).

Links

Crossrefs

Cf. A247049, A247322, A247323, A247325, A247326.

Programs

  • Mathematica
    z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247321 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]]
u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
v = Map[Total, u1]  (* A247322 column sums *)
Table[t[n, 0], {n, 0, z}]   (* A247323, row 0 *)
Table[t[n, 1], {n, 0, z}]   (* A247323 shifted, row 1 *)
Table[t[n, 2], {n, 0, z}]   (* A247325, row 2 *)
Table[t[n, 3], {n, 0, z}]   (* A247326, row 3 *)

Formula

The four rows and the column sums all empirically satisfy the linear recurrence r(n) = 3*r(n-2) + 2*r(n-3) - r(n-4), with g.f. of the form p(x)/q(x), where q(x) = 1 - 3 x^2 - 2 x^3 + x^4. Initial terms and p(x) follow:
(row 0, the bottom row): 1,0,1,1; 1 - 2*x^2 - x^3
(row 1): 0,1,1,2; x + x^2 - x^3
(row 2): 0,1,1,4; x + x^2 + x^3
(row 3): 0,0,1,1; 2x^2 + 2x^3
(n-th column sum) = 1,2,5,9; 1 + 2*x + 2*x^2 + x^3.

A247322 Number of paths from (0,0) to the line x = n, each consisting of segments given by the vectors (1,1), (1,2), (1,-1), with vertices (i,k) satisfying 0 <= k <= 3.

Original entry on oeis.org

1, 2, 5, 9, 18, 35, 67, 132, 253, 495, 956, 1859, 3605, 6994, 13577, 26333, 51114, 99159, 192431, 373372, 724497, 1405819, 2727804, 5293079, 10270553, 19929026, 38670013, 75035105, 145597538, 282516315, 548192811, 1063708916, 2064013525, 4004996055
Offset: 0

Views

Author

Clark Kimberling, Sep 13 2014

Keywords

Comments

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = n-th column sum of the array at A247321.

Examples

			a(2) counts these 5 paths, each represented by a vector sum applied to (0,0): (0,2) + (0,1); (0,1) + (0,2); (0,1) + (0,1); (0,2) + (0,-1), (0,1) + (0,-1).

Links

Crossrefs

Cf. A247049, A247321, A247323, A247325, A247326.

Programs

  • Mathematica
    z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247321 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]]
u1 = Table[t[n, k], {n, 0, z}, {k, 0, 3}];
v = Map[Total, u1]  (* A247322 column sums *)

Formula

A247322(n) = A247323(n) + A247323(n+1) + A247325(n) + A247326(n).
Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (1 + 2*x + 2*x^2 + x^3)/(1 - 3 x^2 - 2 x^3 + x^4).

A247326 Number of paths from (0,0) to (n,3), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 0, 2, 2, 6, 10, 20, 40, 74, 150, 282, 558, 1072, 2088, 4050, 7850, 15254, 29562, 57412, 111344, 216106, 419294, 813594, 1578750, 3063264, 5944144, 11533698, 22380210, 43426118, 84263882, 163505076, 317263672, 615616874, 1194537286, 2317872890, 4497581934
Offset: 0

Views

Author

Clark Kimberling, Sep 13 2014

Keywords

Comments

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 3, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = row 3 of the array at A247321.

Examples

			a(4) counts these 6 paths, each represented by a vector sum applied to (0,0):
(1,2) + (1,1) + (1,-1) + (1,1);
(1,1) + (1,2) + (1,-1) + (1,1);
(1,2) + (1,-1) + (1,1) + (1,1);
(1,1) + (1,-1) + (1,2) + (1,1);
(1,1) + (1,-1) + (1,1) + (1,2);
(1,1) + (1,1) + (1,-1) + (1,2).

Links

Crossrefs

Cf. A247049, A247321, A247322, A247325.

Programs

  • Mathematica
    z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
Table[t[n, 3], {n, 0, z}];  (* A247326 *)

Formula

Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (2*x^2 + x^3)/(1 - 3 x^2 - 2 x^3 + x^4).

A247352 Rectangular array read by upwards by columns: T = T(n,k) = number of paths from (0,1) to (n,k), where 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

0, 1, 0, 0, 1, 0, 1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 2, 5, 8, 5, 5, 10, 12, 13, 10, 17, 28, 22, 17, 38, 49, 45, 38, 66, 100, 87, 66, 138, 191, 166, 138, 257, 370, 329, 257, 508, 724, 627, 508, 981, 1392, 1232, 981, 1900, 2721, 2373, 1900, 3702, 5254, 4621, 3702
Offset: 0

Views

Author

Clark Kimberling, Sep 15 2014

Keywords

Comments

Also, T(n,k) = number of strings s(0)..s(n) of integers such that s(0) = 1, s(n) = k, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n.

Examples

			First 10 columns:
0 .. 1 .. 1 .. 4 .. 5 .. 13 .. 22 .. 45 .. 87 ... 166
0 .. 1 .. 2 .. 3 .. 8 .. 12 .. 28 .. 49 .. 100 .. 191
1 .. 0 .. 2 .. 2 .. 5 .. 10 .. 17 .. 38 .. 66 ... 138
0 .. 1 .. 0 .. 2 .. 2 .. 5 ... 10 .. 17 .. 38 ... 66
T(5,0) counts these 5 paths, given as vector sums applied to (0,1):
(1,1) + (1,1) + (1,-1) + (1,-1) + (1 -1)
(1,1) + (1,-1) + (1,1) + (1,-1) + (1,-1)
(1,-1) + (1,1) + (1,1) + (1,-1) + (1,-1)
(1,1) + (1,-1) + (1,-1) + (1,1) + (1,-1)
(1,-1) + (1,1) + (1,-1) + (1,1) + (1,-1)
Partial sums of second components in each vector sum give the 3 integer strings described in comments:
(1,2,3,2,1,0),
(1,2,1,2,1,0),
(1,0,1,2,1,0),
(1,2,1,0,1,0),
(1,0,1,0,1,0).

Links

Crossrefs

Cf. A247049, A247321, A247353, A247354, A247355.

Programs

  • Mathematica
    z = 50; t[0, 0] = 0; t[0, 1] = 1; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 1; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2]
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3]
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2]
u = Flatten[Table[t[n, k], {n, 0, z}, {k, 0, 3}]] (* A247352 *)
TableForm[Reverse[Transpose[Table[t[n, k], {n, 0, 12}, {k, 0, 3}]]]] (* array  *)
v = Map[Total, u1]  (* A247353 *)
Table[t[n, 0], {n, 0, z}]   (* row 0, A247354*)
Table[t[n, 1], {n, 0, z}]   (* row 1, cf. row 0 *)
Table[t[n, 2], {n, 0, z}]   (* row 2, A247355 *)
Table[t[n, 3], {n, 0, z}]   (* row 3, A247325 *)

Formula

The four rows and column sums all empirically satisfy the linear recurrence r(n) = 3*r(n-2) + 2*r(n-3) - r(n-4), with g.f. of the form p(x)/q(x), where q(x) = 1 - 3 x^2 - 2 x^3 + x^4. Initial terms and p(x) follow:
(row 0, the bottom row): 0,1,0,2; x - x^3
(row 1): 1,0,0,2; 1 - x^2
(row 2): 0,1,2,3; x +2*x^2
(row 3): 0,1,1,4; x + x^2 + x^3
(n-th column sum) = 1,3,5,11; 1 + 3*x + 2*x^2.

A247323 Number of paths from (0,0) to (n,0), with vertices (i,k) satisfying 0 <= k <= 3, consisting of segments given by the vectors (1,1), (1,2), (1,-1).

Original entry on oeis.org

1, 0, 1, 1, 2, 5, 7, 18, 29, 63, 116, 229, 445, 856, 1677, 3229, 6298, 12185, 23675, 45922, 89097, 172931, 335460, 651065, 1263145, 2451184, 4756105, 9228777, 17907538, 34747357, 67424063, 130828370, 253859365, 492585879, 955810772, 1854647997, 3598744709
Offset: 0

Views

Author

Clark Kimberling, Sep 13 2014

Keywords

Comments

Also, a(n) = number of strings s(0)..s(n) of integers such that s(0) = 0, s(n) = 0, and for i > 0, s(i) is in {0,1,2,3} and s(i) - s(i-1) is in {-1,1,2} for 1 <= i <= n; also, a(n) = row 0 (the bottom row) of the array at A247321, and a(n+1) = row 1 of the same array.

Examples

			a(5) counts these 5 paths, each represented by a vector sum applied to (0,0):
(1,2) + (1,1) + (1,-1) + (1,-1) + (1,-1);
(1,1) + (1,2) + (1,-1) + (1,-1) + (1,-1);
(1,2) + (1,-1) + (1,1) + (1,-1) + (1,-1);
(1,1) + (1,-1) + (1,2) + (1,-1) + (1,-1);
(1,2) + (1,-1) + (1,-1) + (1,1) + (1,-1).

Links

Crossrefs

Cf. A247049, A247321, A247322, A247325, A247326.

Programs

  • Mathematica
    z = 25; t[0, 0] = 1; t[0, 1] = 0; t[0, 2] = 0; t[0, 3] = 0;
t[1, 3] = 0; t[n_, 0] := t[n, 0] = t[n - 1, 1];
t[n_, 1] := t[n, 1] = t[n - 1, 0] + t[n - 1, 2];
t[n_, 2] := t[n, 2] = t[n - 1, 0] + t[n - 1, 1] + t[n - 1, 3];
t[n_, 3] := t[n, 3] = t[n - 1, 1] + t[n - 1, 2];
Table[t[n, 0], {n, 0, z}];  (* A247323 *)

Formula

Empirically, a(n) = 3*a(n-2) + 2*a(n-3) - a(n-4) and g.f. = (1 + 2*x^2 - x^3)/(1 - 3 x^2 - 2 x^3 + x^4).
Showing 1-5 of 5 results.

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